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Asymptotic normality determined by high moments, and submap counts of random maps
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  • Published: 25 May 2004

Asymptotic normality determined by high moments, and submap counts of random maps

  • Zhicheng Gao1 &
  • Nicholas C. Wormald2 

Probability Theory and Related Fields volume 130, pages 368–376 (2004)Cite this article

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  • 15 Citations

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Abstract.

We give a general result showing that the asymptotic behaviour of high moments determines the shape of distributions which are asymptotically normal. Both the factorial and non-factorial (non-central) moments are treated. This differs from the usual moment method in combinatorics, as the expected value may tend to infinity quite rapidly. Applications are given to submap counts in random planar triangulations, where we use a simple argument to asymptotically determine high moments for the number of copies of a given subtriangulation in a random 3-connected planar triangulation. Similar results are also obtained for 2-connected triangulations and quadrangulations with no multiple edges.

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Acknowledgments.

The authors would like to thank an anonymous referee for pointing out how to significantly shorten the proof of Theorems 1 and 2 and incidentally remove the condition σ n ( log 2σ n )=o(μ n ). We would also like to thank Boris Pittel for some suggestions.

Author information

Authors and Affiliations

  1. Faculty of Business Administration, University of Macau, Macau, China

    Zhicheng Gao

  2. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON, Canada, N2L 3G1

    Nicholas C. Wormald

Authors
  1. Zhicheng Gao
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  2. Nicholas C. Wormald
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Corresponding author

Correspondence to Zhicheng Gao.

Additional information

Revised version: 6 February 2004

Research supported by NSERCC and University of Macau.

Research supported by the Australian Research Council and the Canada Research Chairs program. Research carried mainly while the author was at the Department of Mathematics and Statistics, University of Melbourne.

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Cite this article

Gao, Z., Wormald, N. Asymptotic normality determined by high moments, and submap counts of random maps. Probab. Theory Relat. Fields 130, 368–376 (2004). https://doi.org/10.1007/s00440-004-0356-9

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  • Received: 06 March 2002

  • Published: 25 May 2004

  • Issue Date: November 2004

  • DOI: https://doi.org/10.1007/s00440-004-0356-9

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Keywords

  • Asymptotic Behaviour
  • General Result
  • Asymptotic Normality
  • High Moment
  • Moment Method
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